Download Congruent Triangles

Geometry Chapter 4
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When you finish the test, please pick up a set
of 9 index cards. Copy the nine theorems and
postulates from chapter four onto these index
cards – including the drawings with them.
Write the name of each theorem or postulate
on the back of the card.
see pages: 199, 205, 206, 213, 214, 228, 235
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2.0 Students write geometric proofs.
4.0 Students prove basic theorems involving
congruence.
5.0 Students prove that triangles are congruent,
and are able to use the concept of corresponding
parts of congruent triangles.
6.0 Students know and are able to use the
triangle inequality theorem.
12.0 Students find and use measures of sides
and of interior and exterior angles of triangles
and polygons to classify figures and solve
problems.
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What do you think makes figures congruent?
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They have the same size and shape.
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If you can slide, flip or turn a shape so that it fits
exactly on another shape, then they are congruent.
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Congruent polygons have congruent corresponding
parts – their matching sides and angles.
Matching vertices are corresponding vertices.
When you name congruent polygons, always list
corresponding vertices in the same order.
Two triangles are congruent if they have three
pairs of congruent corresponding sides, and
three pairs of congruent corresponding angles.
Are the following triangles congruent? Justify
your answer.
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Theorem 4-1: If two angles of one triangle
are congruent to two angles of another
triangle, then the third angles are congruent.
If you can prove that all sides of two triangles
are congruent, then you know the triangles
are congruent.
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The congruent angle must be the INCLUDED
angle between the two sides.
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Warm Up:
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What does the SAS Postulate say about
triangle congruency?
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At your table, choose any two angle measures
that add up to less than 120°. (No zeros)
Agree on a segment length between 5 and 20
centimeters.
Each of you: Draw the line segment, then
construct the given angles on each end of the
segment to form a triangle.
Measure the two remaining sides and
compare your answers.
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What happened?
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Which triangles are congruent?
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homework:
page 215 (1-15) all
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Retake for Chapter 3 test:
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◦ Pick up a retake practice packet.
◦ Complete test corrections.
◦ You MUST have all chapter 3 homework completed
and come in for at least 1 enrichment period before
the retake next Thursday.
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warm up
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Once you show that triangles are congruent
using SSS, SAS, ASA or AAS, then you can
make conclusions about the other parts of
the triangles because, by definition,
congruent parts of congruent triangles are
congruent.
Abbreviate this CPCTC
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Before you can use CPCTC in a proof, you
must first show that the triangles are
congruent.
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Warm Up
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Construct an Isosceles Triangle
1. Use a straight edge to make a line segment. Label
the endpoints A and B.
2. Set your compass to a length that is greater than
half the length of the segment.
3. Without changing the compass setting, make arcs
from either end of the line segment.
4. Connect the endpoints of the segment to the
intersection point of the two arcs. Label this point C.
5. Measure the sides of the triangle to confirm that
they are equal.
4-5 Isosceles and Equilateral Triangles
EQ: How do you use the properties of Isosceles triangles in proofs?
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Fold your triangle carefully in half, so points
A and B are exactly on top of each other.
Label the point where the fold intersects AB
as point D.
What appears to be true of angles A and B?
What appears to be true of the intersection of
CD and AB?
Write a conjecture about the angles opposite
the congruent sides of an isosceles triangle.
4-5 Isosceles and Equilateral Triangles
EQ: How do you use the properties of Isosceles triangles in proofs?
4-5 Isosceles and Equilateral Triangles
EQ: How do you use the properties of Isosceles triangles in proofs?
4-5 Isosceles and Equilateral Triangles
EQ: How do you use the properties of Isosceles triangles in proofs?
4-5 Isosceles and Equilateral Triangles
EQ: How do you use the properties of Isosceles triangles in proofs?
4-5 Isosceles and Equilateral Triangles
EQ: How do you use the properties of Isosceles triangles in proofs?
A corollary is a statement that follows directly
from a theorem
4-5 Isosceles and Equilateral Triangles
EQ: How do you use the properties of Isosceles triangles in proofs?
4-5 Isosceles and Equilateral Triangles
EQ: How do you use the properties of Isosceles triangles in proofs?
4-5 Isosceles and Equilateral Triangles
EQ: How do you use the properties of Isosceles triangles in proofs?
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Warm Up
Homework: p237 (1-8)
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Warm Up:
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When a geometric drawing is complicated, it
is sometimes helpful to separate it into more
than one drawing.
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Sometimes you can prove triangles are
congruent and then use their corresponding
parts to prove another pair congruent.
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Worksheet 4-7, both sides
Chapter 4 test Tuesday – period 3
Chapter 4 test Wednesday – period 6
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h
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i
g
j
Draw RSTU
congruent to GHIJ.
List all the
congruent parts of
the two figures.
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What else would you need to have to prove
these triangles congruent by SSS?
By SAS?
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What other piece of
information do you need
to prove these triangles
are congruent?
By ASA?
By SAS?
By AAS?
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prove ∠P ≅∠Q
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prove ∠P ≅∠Q
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prove ∠P ≅∠Q
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Given: KM≅LJ, KJ ≅ LM
Prove: OJ ≅ OM